Bilinear Fourier restriction theorems

TL;DR

This paper introduces a new method for establishing $L^p$ estimates for bilinear Fourier restriction problems, applying it to the lacunary polygon without relying on classical Littlewood--Paley techniques.

Contribution

It presents a novel scheme for bilinear Fourier restriction estimates that bypasses traditional Littlewood--Paley inequalities, expanding the tools available for such problems.

Findings

Established $L^p$ estimates for the lacunary polygon

Developed a general scheme applicable to bilinear Fourier restrictions

Avoided use of Rubio de Francia Littlewood--Paley inequality

Abstract

We provide a general scheme for proving $L^p$ estimates for certain bilinear Fourier restrictions outside the locally $L^2$ setting. As an application, we show how such estimates follow for the lacunary polygon. In contrast with prior approaches, our argument avoids any use of the Rubio de Francia Littlewood--Paley inequality.